Random graphs with arbitrary degree distributions and their applications

TLDR

Recent work on the structure of social networks and the internet has focused attention on graphs with degree distributions that differ markedly from the Poisson distributions previously studied. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. The authors examine directed, bipartite, and undirected graphs and apply the theory to real‑world networks such as the world‑wide web and collaboration graphs of scientists and Fortune 1000 directors. The study derives exact expressions for the phase transition, component sizes, and average distances, and shows that while the theory predicts real‑world network behavior accurately in some cases, discrepancies in others suggest additional social structure beyond random graph assumptions.

Abstract

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.